A weighted sum over generalized Tesler matrices

作者：Wilson Andrew Timothy^*

来源：Journal of Algebraic Combinatorics, 2017, 45(3): 825-855.

DOI：10.1007/s10801-016-0726-2

• Tesler matrices
• Macdonald polynomials
• Shuffle Conjecture
• Ordered set partitions
• Parking functions

We generalize previous definitions of Tesler matrices to allow negative matrix entries and negative hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these matrices, which we call the Tesler function. Our interpretation uses a new class of symmetric function specializations which are defined by their values on Macdonald polynomials. As a result of this interpretation, we obtain a Tesler function expression for the Hall inner product , where is the delta operator introduced by Bergeron, Garsia, Haiman, and Tesler. We also provide simple formulas for various special cases of Tesler functions which involve q, t-binomial coefficients, ordered set partitions, and parking functions. These formulas prove two cases of the recent Delta Conjecture posed by Haglund, Remmel, and the author.

• 出版日期2017-5